Reciprocal lattice and Ewalds sphere

The better visual representation of the phenomenon of diffraction has been introduced by Ewald (see the footnote on page 50). Consider an incident wave with a certain propagation vector, ko, and a wavelength, X. If the length of ko is selected as the inverse of the wavelength 

Its length is known (1/A.) and its orientation, i.e. angle 0, is found by a simple algebraic transformation after recalling that d = Md  

The Ewald s sphere and the reciprocal lattice are essential tools in the visualization of the three-dimensional diffraction patterns from single crystals, as will be illustrated in the next few paragraphs. They are also invaluable in the understanding of the geometry of diffraction from polycrystalline (powder) specimens, which will be explained in the next section. 

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Figure 5.9. (a) Geometry of a grain boundary normal to the plane of the foil and parallel to the direction m of the incident electron beam. The boundary has a periodicity along the direction p, and the boundary normal is specified by n. (b) The reciprocal lattice and Ewald sphere in the neighborhood of a matrix reflection, showing the extra reflections due to the periodic boundary and the streaking of the reflections in the direction n. (After Carter and Sass 1981.)... 

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